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Description:
Assume n≥2. Consider the elementary symmetric polynomials ek(y1,y2,…,yn) and denote by E0,E1,…,En−1 the elementary symmetric polynomials in reverse order Ek(y1,y2,…,yn):=en−k(y1,y2,…,yn)=∑i1<⋯<in−kyi1yi2⋯yin−k, k∈{0,1,…,n−1}. Let, moreover, S be a nonempty subset of {0,1,…,n−1}. We investigate necessary and sufficient conditions on the function f:I→R, where I⊂R is an interval, such that the inequality f(a1)+f(a2)+⋯+f(an)≤f(b1)+f(b2)+⋯+f(bn) (∗) holds for all a=(a1,a2,…,an)∈In and b=(b1,b2,…,bn)∈In satisfying Ek(a)<Ek(b) for k∈S and Ek(a)=Ek(b) for k∈{0,1,…,n−1}∖S. As a corollary, we obtain our inequality (∗) if 2≤n≤4, f(x)=log2x and S={1,…,n−1}, which is the sum of squared logarithms inequality previously known for 2≤n≤3.